\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  18.0 & +  7.00 x_{1} & -4.00 x_{2} & +  3.00 x_{3} & -7.00 x_{4} & -10.00 x_{5} & -2.00 x_{6} & -3.00 x_{7} & -10.00 x_{8}\\
 x_{10}   &  0.0 & +  4.00 x_{1} & -5.00 x_{2} & -7.00 x_{3} & +  2.00 x_{4} & -9.00 x_{5} & -3.00 x_{6} & +  6.00 x_{7} & +  3.00 x_{8}\\
 x_{11}   &  1.0 & +  4.00 x_{1} & -6.00 x_{2} & -9.00 x_{3} & -2.00 x_{4} & +  4.00 x_{5} & -7.00 x_{6} & +  9.00 x_{7} & -4.00 x_{8}\\
 x_{12}   &  51.0 & +  3.00 x_{1} & -7.00 x_{2} & -4.00 x_{3} & -4.00 x_{4} & +  8.00 x_{5} & -3.00 x_{6} & -9.00 x_{7} & -8.00 x_{8}\\
 x_{13}   &  -4.0 & + 10.00 x_{1} & -7.00 x_{2} &   & +  2.00 x_{4} & +  3.00 x_{5} & +  6.00 x_{6} & +  4.00 x_{7} & +  5.00 x_{8}\\
 x_{14}   &  5.0 & +  7.00 x_{1} & +  7.00 x_{2} & -8.00 x_{3} & -3.00 x_{4} & +  4.00 x_{5} & -3.00 x_{6} & +  5.00 x_{7} & -9.00 x_{8}\\
\hline
z    &  0.0 & -5.00 x_{1} & -2.00 x_{2} & -1.00 x_{3} & +  3.00 x_{4} & +  3.00 x_{5} & +  1.00 x_{6} & +  5.00 x_{7} &   \\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,14$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -7.00 y_{9} & -4.00 y_{10} & -4.00 y_{11} & -3.00 y_{12} & -10.00 y_{13} & -7.00 y_{14}\\
 y_{2}   &  1.0 & +  4.00 y_{9} & +  5.00 y_{10} & +  6.00 y_{11} & +  7.00 y_{12} & +  7.00 y_{13} & -7.00 y_{14}\\
 y_{3}   &  1.0 & -3.00 y_{9} & +  7.00 y_{10} & +  9.00 y_{11} & +  4.00 y_{12} &   & +  8.00 y_{14}\\
 y_{4}   &  1.0 & +  7.00 y_{9} & -2.00 y_{10} & +  2.00 y_{11} & +  4.00 y_{12} & -2.00 y_{13} & +  3.00 y_{14}\\
 y_{5}   &  1.0 & + 10.00 y_{9} & +  9.00 y_{10} & -4.00 y_{11} & -8.00 y_{12} & -3.00 y_{13} & -4.00 y_{14}\\
 y_{6}   &  1.0 & +  2.00 y_{9} & +  3.00 y_{10} & +  7.00 y_{11} & +  3.00 y_{12} & -6.00 y_{13} & +  3.00 y_{14}\\
 y_{7}   &  1.0 & +  3.00 y_{9} & -6.00 y_{10} & -9.00 y_{11} & +  9.00 y_{12} & -4.00 y_{13} & -5.00 y_{14}\\
 y_{8}   &  1.0 & + 10.00 y_{9} & -3.00 y_{10} & +  4.00 y_{11} & +  8.00 y_{12} & -5.00 y_{13} & +  9.00 y_{14}\\
\hline
z    &  -0 & -18.00 y_{9} &   & -1.00 y_{11} & -51.00 y_{12} & +  4.00 y_{13} & -5.00 y_{14}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 2 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{13}   &  0.1 & -0.70 y_{9} & -0.40 y_{10} & -0.40 y_{11} & -0.30 y_{12} & -0.10 y_{1} & -0.70 y_{14}\\
 y_{2}   &  1.7 & -0.90 y_{9} & +  2.20 y_{10} & +  3.20 y_{11} & +  4.90 y_{12} & -0.70 y_{1} & -11.90 y_{14}\\
 y_{3}   &  1.0 & -3.00 y_{9} & +  7.00 y_{10} & +  9.00 y_{11} & +  4.00 y_{12} &   & +  8.00 y_{14}\\
 y_{4}   &  0.8 & +  8.40 y_{9} & -1.20 y_{10} & +  2.80 y_{11} & +  4.60 y_{12} & +  0.20 y_{1} & +  4.40 y_{14}\\
 y_{5}   &  0.7 & + 12.10 y_{9} & + 10.20 y_{10} & -2.80 y_{11} & -7.10 y_{12} & +  0.30 y_{1} & -1.90 y_{14}\\
 y_{6}   &  0.4 & +  6.20 y_{9} & +  5.40 y_{10} & +  9.40 y_{11} & +  4.80 y_{12} & +  0.60 y_{1} & +  7.20 y_{14}\\
 y_{7}   &  0.6 & +  5.80 y_{9} & -4.40 y_{10} & -7.40 y_{11} & + 10.20 y_{12} & +  0.40 y_{1} & -2.20 y_{14}\\
 y_{8}   &  0.5 & + 13.50 y_{9} & -1.00 y_{10} & +  6.00 y_{11} & +  9.50 y_{12} & +  0.50 y_{1} & + 12.50 y_{14}\\
\hline
z    &  0.4 & -20.80 y_{9} & -1.60 y_{10} & -2.60 y_{11} & -52.20 y_{12} & -0.40 y_{1} & -7.80 y_{14}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  20.8 & +  0.70 x_{13} & +  0.90 x_{2} & +  3.00 x_{3} & -8.40 x_{4} & -12.10 x_{5} & -6.20 x_{6} & -5.80 x_{7} & -13.50 x_{8}\\
 x_{10}   &  1.6 & +  0.40 x_{13} & -2.20 x_{2} & -7.00 x_{3} & +  1.20 x_{4} & -10.20 x_{5} & -5.40 x_{6} & +  4.40 x_{7} & +  1.00 x_{8}\\
 x_{11}   &  2.6 & +  0.40 x_{13} & -3.20 x_{2} & -9.00 x_{3} & -2.80 x_{4} & +  2.80 x_{5} & -9.40 x_{6} & +  7.40 x_{7} & -6.00 x_{8}\\
 x_{12}   &  52.2 & +  0.30 x_{13} & -4.90 x_{2} & -4.00 x_{3} & -4.60 x_{4} & +  7.10 x_{5} & -4.80 x_{6} & -10.20 x_{7} & -9.50 x_{8}\\
 x_{1}   &  0.4 & +  0.10 x_{13} & +  0.70 x_{2} &   & -0.20 x_{4} & -0.30 x_{5} & -0.60 x_{6} & -0.40 x_{7} & -0.50 x_{8}\\
 x_{14}   &  7.8 & +  0.70 x_{13} & + 11.90 x_{2} & -8.00 x_{3} & -4.40 x_{4} & +  1.90 x_{5} & -7.20 x_{6} & +  2.20 x_{7} & -12.50 x_{8}\\
\hline
z    &  -0.4 & -0.10 x_{13} & -1.70 x_{2} & -1.00 x_{3} & -0.80 x_{4} & -0.70 x_{5} & -0.40 x_{6} & -0.60 x_{7} & -0.50 x_{8}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  20.8 & +  0.70 x_{13} & +  0.90 x_{2} & +  3.00 x_{3} & -8.40 x_{4} & -12.10 x_{5} & -6.20 x_{6} & -5.80 x_{7} & -13.50 x_{8}\\
 x_{10}   &  1.6 & +  0.40 x_{13} & -2.20 x_{2} & -7.00 x_{3} & +  1.20 x_{4} & -10.20 x_{5} & -5.40 x_{6} & +  4.40 x_{7} & +  1.00 x_{8}\\
 x_{11}   &  2.6 & +  0.40 x_{13} & -3.20 x_{2} & -9.00 x_{3} & -2.80 x_{4} & +  2.80 x_{5} & -9.40 x_{6} & +  7.40 x_{7} & -6.00 x_{8}\\
 x_{12}   &  52.2 & +  0.30 x_{13} & -4.90 x_{2} & -4.00 x_{3} & -4.60 x_{4} & +  7.10 x_{5} & -4.80 x_{6} & -10.20 x_{7} & -9.50 x_{8}\\
 x_{1}   &  0.4 & +  0.10 x_{13} & +  0.70 x_{2} &   & -0.20 x_{4} & -0.30 x_{5} & -0.60 x_{6} & -0.40 x_{7} & -0.50 x_{8}\\
 x_{14}   &  7.8 & +  0.70 x_{13} & + 11.90 x_{2} & -8.00 x_{3} & -4.40 x_{4} & +  1.90 x_{5} & -7.20 x_{6} & +  2.20 x_{7} & -12.50 x_{8}\\
\hline
z    &  -2.0 & -0.50 x_{13} & -5.50 x_{2} & -1.00 x_{3} & +  4.00 x_{4} & +  4.50 x_{5} & +  4.00 x_{6} & +  7.00 x_{7} & +  2.50 x_{8}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  20.8 & +  0.70 x_{13} & +  0.90 x_{2} & +  3.00 x_{3} & -8.40 x_{4} & -12.10 x_{5} & -6.20 x_{6} & -5.80 x_{7} & -13.50 x_{8}\\
 x_{10}   &  1.6 & +  0.40 x_{13} & -2.20 x_{2} & -7.00 x_{3} & +  1.20 x_{4} & -10.20 x_{5} & -5.40 x_{6} & +  4.40 x_{7} & +  1.00 x_{8}\\
 x_{11}   &  2.6 & +  0.40 x_{13} & -3.20 x_{2} & -9.00 x_{3} & -2.80 x_{4} & +  2.80 x_{5} & -9.40 x_{6} & +  7.40 x_{7} & -6.00 x_{8}\\
 x_{12}   &  52.2 & +  0.30 x_{13} & -4.90 x_{2} & -4.00 x_{3} & -4.60 x_{4} & +  7.10 x_{5} & -4.80 x_{6} & -10.20 x_{7} & -9.50 x_{8}\\
 x_{1}   &  0.4 & +  0.10 x_{13} & +  0.70 x_{2} &   & -0.20 x_{4} & -0.30 x_{5} & -0.60 x_{6} & -0.40 x_{7} & -0.50 x_{8}\\
 x_{14}   &  7.8 & +  0.70 x_{13} & + 11.90 x_{2} & -8.00 x_{3} & -4.40 x_{4} & +  1.90 x_{5} & -7.20 x_{6} & +  2.20 x_{7} & -12.50 x_{8}\\
\hline
z    &  -2.0 & -0.50 x_{13} & -5.50 x_{2} & -1.00 x_{3} & +  4.00 x_{4} & +  4.50 x_{5} & +  4.00 x_{6} & +  7.00 x_{7} & +  2.50 x_{8}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  13.0 & -0.50 x_{13} & + 10.50 x_{2} & + 30.00 x_{3} & +  3.00 x_{11} & -20.50 x_{5} & + 22.00 x_{6} & -28.00 x_{7} & +  4.50 x_{8}\\
 x_{10}   &  2.71428571429 & +  0.57 x_{13} & -3.57 x_{2} & -10.86 x_{3} & -0.43 x_{11} & -9.00 x_{5} & -9.43 x_{6} & +  7.57 x_{7} & -1.57 x_{8}\\
 x_{4}   &  0.928571428571 & +  0.14 x_{13} & -1.14 x_{2} & -3.21 x_{3} & -0.36 x_{11} & +  1.00 x_{5} & -3.36 x_{6} & +  2.64 x_{7} & -2.14 x_{8}\\
 x_{12}   &  47.9285714286 & -0.36 x_{13} & +  0.36 x_{2} & + 10.79 x_{3} & +  1.64 x_{11} & +  2.50 x_{5} & + 10.64 x_{6} & -22.36 x_{7} & +  0.36 x_{8}\\
 x_{1}   &  0.214285714286 & +  0.07 x_{13} & +  0.93 x_{2} & +  0.64 x_{3} & +  0.07 x_{11} & -0.50 x_{5} & +  0.07 x_{6} & -0.93 x_{7} & -0.07 x_{8}\\
 x_{14}   &  3.71428571429 & +  0.07 x_{13} & + 16.93 x_{2} & +  6.14 x_{3} & +  1.57 x_{11} & -2.50 x_{5} & +  7.57 x_{6} & -9.43 x_{7} & -3.07 x_{8}\\
\hline
z    &  1.71428571429 & +  0.07 x_{13} & -10.07 x_{2} & -13.86 x_{3} & -1.43 x_{11} & +  8.50 x_{5} & -9.43 x_{6} & + 17.57 x_{7} & -6.07 x_{8}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  6.81746031746 & -1.80 x_{13} & + 18.63 x_{2} & + 54.73 x_{3} & +  3.98 x_{11} & +  2.28 x_{10} & + 43.48 x_{6} & -45.25 x_{7} & +  8.08 x_{8}\\
 x_{5}   &  0.301587301587 & +  0.06 x_{13} & -0.40 x_{2} & -1.21 x_{3} & -0.05 x_{11} & -0.11 x_{10} & -1.05 x_{6} & +  0.84 x_{7} & -0.17 x_{8}\\
 x_{4}   &  1.23015873016 & +  0.21 x_{13} & -1.54 x_{2} & -4.42 x_{3} & -0.40 x_{11} & -0.11 x_{10} & -4.40 x_{6} & +  3.48 x_{7} & -2.32 x_{8}\\
 x_{12}   &  48.6825396825 & -0.20 x_{13} & -0.63 x_{2} & +  7.77 x_{3} & +  1.52 x_{11} & -0.28 x_{10} & +  8.02 x_{6} & -20.25 x_{7} & -0.08 x_{8}\\
 x_{1}   &  0.0634920634921 & +  0.04 x_{13} & +  1.13 x_{2} & +  1.25 x_{3} & +  0.10 x_{11} & +  0.06 x_{10} & +  0.60 x_{6} & -1.35 x_{7} & +  0.02 x_{8}\\
 x_{14}   &  2.96031746032 & -0.09 x_{13} & + 17.92 x_{2} & +  9.16 x_{3} & +  1.69 x_{11} & +  0.28 x_{10} & + 10.19 x_{6} & -11.53 x_{7} & -2.63 x_{8}\\
\hline
z    &  4.27777777778 & +  0.61 x_{13} & -13.44 x_{2} & -24.11 x_{3} & -1.83 x_{11} & -0.94 x_{10} & -18.33 x_{6} & + 24.72 x_{7} & -7.56 x_{8}\\
\end{array}\]


 $ x_{7} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  4.68823529412 & -3.13 x_{13} & -19.16 x_{2} & + 12.94 x_{3} & +  0.78 x_{11} & +  0.41 x_{10} & + 23.51 x_{6} & + 33.54 x_{1} & +  7.55 x_{8}\\
 x_{5}   &  0.341176470588 & +  0.09 x_{13} & +  0.31 x_{2} & -0.43 x_{3} & +  0.01 x_{11} & -0.08 x_{10} & -0.68 x_{6} & -0.62 x_{1} & -0.16 x_{8}\\
 x_{4}   &  1.39411764706 & +  0.31 x_{13} & +  1.37 x_{2} & -1.20 x_{3} & -0.16 x_{11} & +  0.03 x_{10} & -2.87 x_{6} & -2.58 x_{1} & -2.28 x_{8}\\
 x_{12}   &  47.7294117647 & -0.79 x_{13} & -17.55 x_{2} & -10.94 x_{3} & +  0.09 x_{11} & -1.11 x_{10} & -0.91 x_{6} & + 15.01 x_{1} & -0.32 x_{8}\\
 x_{7}   &  0.0470588235294 & +  0.03 x_{13} & +  0.84 x_{2} & +  0.92 x_{3} & +  0.07 x_{11} & +  0.04 x_{10} & +  0.44 x_{6} & -0.74 x_{1} & +  0.01 x_{8}\\
 x_{14}   &  2.41764705882 & -0.43 x_{13} & +  8.29 x_{2} & -1.49 x_{3} & +  0.88 x_{11} & -0.20 x_{10} & +  5.10 x_{6} & +  8.55 x_{1} & -2.77 x_{8}\\
\hline
z    &  5.44117647059 & +  1.34 x_{13} & +  7.21 x_{2} & -1.28 x_{3} & -0.09 x_{11} & +  0.07 x_{10} & -7.43 x_{6} & -18.32 x_{1} & -7.26 x_{8}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  0.244703715075 & -0.16 x_{13} & -0.05 x_{9} & +  0.68 x_{3} & +  0.04 x_{11} & +  0.02 x_{10} & +  1.23 x_{6} & +  1.75 x_{1} & +  0.39 x_{8}\\
 x_{5}   &  0.416027018729 & +  0.04 x_{13} & -0.02 x_{9} & -0.22 x_{3} & +  0.02 x_{11} & -0.07 x_{10} & -0.30 x_{6} & -0.09 x_{1} & -0.04 x_{8}\\
 x_{4}   &  1.72950568007 & +  0.08 x_{13} & -0.07 x_{9} & -0.28 x_{3} & -0.10 x_{11} & +  0.06 x_{10} & -1.19 x_{6} & -0.18 x_{1} & -1.74 x_{8}\\
 x_{12}   &  43.4341418483 & +  2.08 x_{13} & +  0.92 x_{9} & -22.79 x_{3} & -0.62 x_{11} & -1.49 x_{10} & -22.46 x_{6} & -15.71 x_{1} & -7.23 x_{8}\\
 x_{7}   &  0.251458397298 & -0.11 x_{13} & -0.04 x_{9} & +  1.49 x_{3} & +  0.10 x_{11} & +  0.06 x_{10} & +  1.47 x_{6} & +  0.72 x_{1} & +  0.34 x_{8}\\
 x_{14}   &  4.44580902671 & -1.78 x_{13} & -0.43 x_{9} & +  4.11 x_{3} & +  1.21 x_{11} & -0.02 x_{10} & + 15.28 x_{6} & + 23.05 x_{1} & +  0.49 x_{8}\\
\hline
z    &  7.20448265275 & +  0.16 x_{13} & -0.38 x_{9} & +  3.59 x_{3} & +  0.21 x_{11} & +  0.23 x_{10} & +  1.42 x_{6} & -5.71 x_{1} & -4.43 x_{8}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  1.50654720882 & -0.05 x_{13} & -0.10 x_{9} & -3.03 x_{5} & +  0.11 x_{11} & -0.19 x_{10} & +  0.31 x_{6} & +  1.48 x_{1} & +  0.26 x_{8}\\
 x_{3}   &  1.86767746382 & +  0.17 x_{13} & -0.07 x_{9} & -4.49 x_{5} & +  0.11 x_{11} & -0.31 x_{10} & -1.35 x_{6} & -0.40 x_{1} & -0.20 x_{8}\\
 x_{4}   &  1.21226740179 & +  0.04 x_{13} & -0.05 x_{9} & +  1.24 x_{5} & -0.13 x_{11} & +  0.15 x_{10} & -0.81 x_{6} & -0.07 x_{1} & -1.68 x_{8}\\
 x_{12}   &  0.861474844934 & -1.84 x_{13} & +  2.55 x_{9} & + 102.33 x_{5} & -3.10 x_{11} & +  5.66 x_{10} & +  8.35 x_{6} & -6.70 x_{1} & -2.71 x_{8}\\
 x_{7}   &  3.03032391454 & +  0.15 x_{13} & -0.15 x_{9} & -6.68 x_{5} & +  0.27 x_{11} & -0.41 x_{10} & -0.54 x_{6} & +  0.13 x_{1} & +  0.05 x_{8}\\
 x_{14}   &  12.1192281185 & -1.08 x_{13} & -0.73 x_{9} & -18.44 x_{5} & +  1.66 x_{11} & -1.31 x_{10} & +  9.72 x_{6} & + 21.43 x_{1} & -0.32 x_{8}\\
\hline
z    &  13.9076498966 & +  0.78 x_{13} & -0.63 x_{9} & -16.11 x_{5} & +  0.60 x_{11} & -0.90 x_{10} & -3.43 x_{6} & -7.13 x_{1} & -5.14 x_{8}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  1.53829078801 & -0.12 x_{13} & -0.01 x_{9} & +  0.74 x_{5} & -0.04 x_{12} & +  0.02 x_{10} & +  0.62 x_{6} & +  1.24 x_{1} & +  0.16 x_{8}\\
 x_{3}   &  1.89789123196 & +  0.11 x_{13} & +  0.02 x_{9} & -0.90 x_{5} & -0.04 x_{12} & -0.12 x_{10} & -1.06 x_{6} & -0.63 x_{1} & -0.29 x_{8}\\
 x_{4}   &  1.17536071032 & +  0.12 x_{13} & -0.16 x_{9} & -3.14 x_{5} & +  0.04 x_{12} & -0.09 x_{10} & -1.17 x_{6} & +  0.21 x_{1} & -1.57 x_{8}\\
 x_{11}   &  0.277469478357 & -0.59 x_{13} & +  0.82 x_{9} & + 32.96 x_{5} & -0.32 x_{12} & +  1.82 x_{10} & +  2.69 x_{6} & -2.16 x_{1} & -0.87 x_{8}\\
 x_{7}   &  3.10432852386 & -0.01 x_{13} & +  0.07 x_{9} & +  2.11 x_{5} & -0.09 x_{12} & +  0.08 x_{10} & +  0.17 x_{6} & -0.44 x_{1} & -0.19 x_{8}\\
 x_{14}   &  12.5804661487 & -2.06 x_{13} & +  0.64 x_{9} & + 36.34 x_{5} & -0.54 x_{12} & +  1.72 x_{10} & + 14.19 x_{6} & + 17.84 x_{1} & -1.77 x_{8}\\
\hline
z    &  14.0732519423 & +  0.42 x_{13} & -0.14 x_{9} & +  3.56 x_{5} & -0.19 x_{12} & +  0.19 x_{10} & -1.83 x_{6} & -8.42 x_{1} & -5.66 x_{8}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  1.81433316842 & -0.09 x_{13} & -0.04 x_{9} & -0.23 x_{4} & -0.03 x_{12} & -0.00 x_{10} & +  0.35 x_{6} & +  1.29 x_{1} & -0.21 x_{8}\\
 x_{3}   &  1.56095837162 & +  0.07 x_{13} & +  0.06 x_{9} & +  0.29 x_{4} & -0.05 x_{12} & -0.09 x_{10} & -0.72 x_{6} & -0.69 x_{1} & +  0.16 x_{8}\\
 x_{5}   &  0.374231394445 & +  0.04 x_{13} & -0.05 x_{9} & -0.32 x_{4} & +  0.01 x_{12} & -0.03 x_{10} & -0.37 x_{6} & +  0.07 x_{1} & -0.50 x_{8}\\
 x_{11}   &  12.6119867128 & +  0.62 x_{13} & -0.87 x_{9} & -10.49 x_{4} & +  0.13 x_{12} & +  0.84 x_{10} & -9.58 x_{6} & +  0.08 x_{1} & -17.30 x_{8}\\
 x_{7}   &  3.8944094989 & +  0.07 x_{13} & -0.04 x_{9} & -0.67 x_{4} & -0.06 x_{12} & +  0.02 x_{10} & -0.61 x_{6} & -0.30 x_{1} & -1.24 x_{8}\\
 x_{14}   &  26.1816382783 & -0.72 x_{13} & -1.22 x_{9} & -11.57 x_{4} & -0.04 x_{12} & +  0.64 x_{10} & +  0.67 x_{6} & + 20.31 x_{1} & -19.89 x_{8}\\
\hline
z    &  15.4051169694 & +  0.55 x_{13} & -0.33 x_{9} & -1.13 x_{4} & -0.14 x_{12} & +  0.09 x_{10} & -3.15 x_{6} & -8.18 x_{1} & -7.43 x_{8}\\
\end{array}\]


 $ x_{10} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  1.76722090261 & -0.09 x_{13} & -0.04 x_{9} & -0.19 x_{4} & -0.03 x_{12} & +  0.13 x_{5} & +  0.39 x_{6} & +  1.28 x_{1} & -0.14 x_{8}\\
 x_{3}   &  0.44893111639 & -0.04 x_{13} & +  0.22 x_{9} & +  1.23 x_{4} & -0.09 x_{12} & +  2.97 x_{5} & +  0.38 x_{6} & -0.89 x_{1} & +  1.64 x_{8}\\
 x_{10}   &  12.5771971496 & +  1.24 x_{13} & -1.72 x_{9} & -10.70 x_{4} & +  0.46 x_{12} & -33.61 x_{5} & -12.51 x_{6} & +  2.28 x_{1} & -16.75 x_{8}\\
 x_{11}   &  23.190023753 & +  1.67 x_{13} & -2.32 x_{9} & -19.49 x_{4} & +  0.51 x_{12} & -28.27 x_{5} & -20.10 x_{6} & +  2.00 x_{1} & -31.39 x_{8}\\
 x_{7}   &  4.09263657957 & +  0.09 x_{13} & -0.07 x_{9} & -0.84 x_{4} & -0.05 x_{12} & -0.53 x_{5} & -0.81 x_{6} & -0.26 x_{1} & -1.50 x_{8}\\
 x_{14}   &  34.242280285 & +  0.08 x_{13} & -2.33 x_{9} & -18.43 x_{4} & +  0.25 x_{12} & -21.54 x_{5} & -7.35 x_{6} & + 21.77 x_{1} & -30.62 x_{8}\\
\hline
z    &  16.4798099762 & +  0.66 x_{13} & -0.47 x_{9} & -2.05 x_{4} & -0.10 x_{12} & -2.87 x_{5} & -4.22 x_{6} & -7.98 x_{1} & -8.86 x_{8}\\
\end{array}\]


 $ x_{13} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  0.6 & +  2.60 x_{3} & -0.60 x_{9} & -3.40 x_{4} & +  0.20 x_{12} & -7.60 x_{5} & -0.60 x_{6} & +  3.60 x_{1} & -4.40 x_{8}\\
 x_{13}   &  12.6 & -28.07 x_{3} & +  6.07 x_{9} & + 34.60 x_{4} & -2.47 x_{12} & + 83.40 x_{5} & + 10.73 x_{6} & -25.07 x_{1} & + 45.93 x_{8}\\
 x_{10}   &  28.2 & -34.80 x_{3} & +  5.80 x_{9} & + 32.20 x_{4} & -2.60 x_{12} & + 69.80 x_{5} & +  0.80 x_{6} & -28.80 x_{1} & + 40.20 x_{8}\\
 x_{11}   &  44.2 & -46.80 x_{3} & +  7.80 x_{9} & + 38.20 x_{4} & -3.60 x_{12} & + 110.80 x_{5} & -2.20 x_{6} & -39.80 x_{1} & + 45.20 x_{8}\\
 x_{7}   &  5.2 & -2.47 x_{3} & +  0.47 x_{9} & +  2.20 x_{4} & -0.27 x_{12} & +  6.80 x_{5} & +  0.13 x_{6} & -2.47 x_{1} & +  2.53 x_{8}\\
 x_{14}   &  35.2 & -2.13 x_{3} & -1.87 x_{9} & -15.80 x_{4} & +  0.07 x_{12} & -15.20 x_{5} & -6.53 x_{6} & + 19.87 x_{1} & -27.13 x_{8}\\
\hline
z    &  24.8 & -18.53 x_{3} & +  3.53 x_{9} & + 20.80 x_{4} & -1.73 x_{12} & + 52.20 x_{5} & +  2.87 x_{6} & -24.53 x_{1} & + 21.47 x_{8}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.176470588235 & +  0.76 x_{3} & -0.18 x_{9} & -0.29 x_{2} & +  0.06 x_{12} & -2.24 x_{5} & -0.18 x_{6} & +  1.06 x_{1} & -1.29 x_{8}\\
 x_{13}   &  18.7058823529 & -1.61 x_{3} & -0.04 x_{9} & -10.18 x_{2} & -0.43 x_{12} & +  6.06 x_{5} & +  4.63 x_{6} & + 11.57 x_{1} & +  1.16 x_{8}\\
 x_{10}   &  33.8823529412 & -10.18 x_{3} & +  0.12 x_{9} & -9.47 x_{2} & -0.71 x_{12} & -2.18 x_{5} & -4.88 x_{6} & +  5.29 x_{1} & -1.47 x_{8}\\
 x_{11}   &  50.9411764706 & -17.59 x_{3} & +  1.06 x_{9} & -11.24 x_{2} & -1.35 x_{12} & + 25.41 x_{5} & -8.94 x_{6} & +  0.65 x_{1} & -4.24 x_{8}\\
 x_{7}   &  5.58823529412 & -0.78 x_{3} & +  0.08 x_{9} & -0.65 x_{2} & -0.14 x_{12} & +  1.88 x_{5} & -0.25 x_{6} & -0.14 x_{1} & -0.31 x_{8}\\
 x_{14}   &  32.4117647059 & -14.22 x_{3} & +  0.92 x_{9} & +  4.65 x_{2} & -0.86 x_{12} & + 20.12 x_{5} & -3.75 x_{6} & +  3.14 x_{1} & -6.69 x_{8}\\
\hline
z    &  28.4705882353 & -2.63 x_{3} & -0.14 x_{9} & -6.12 x_{2} & -0.51 x_{12} & +  5.71 x_{5} & -0.80 x_{6} & -2.51 x_{1} & -5.45 x_{8}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.0789473684211 & +  0.34 x_{3} & -0.08 x_{9} & -0.13 x_{2} & +  0.03 x_{12} & -0.45 x_{4} & -0.08 x_{6} & +  0.47 x_{1} & -0.58 x_{8}\\
 x_{13}   &  19.1842105263 & +  0.46 x_{3} & -0.52 x_{9} & -10.97 x_{2} & -0.27 x_{12} & -2.71 x_{4} & +  4.15 x_{6} & + 14.44 x_{1} & -2.35 x_{8}\\
 x_{10}   &  33.7105263158 & -10.92 x_{3} & +  0.29 x_{9} & -9.18 x_{2} & -0.76 x_{12} & +  0.97 x_{4} & -4.71 x_{6} & +  4.26 x_{1} & -0.21 x_{8}\\
 x_{11}   &  52.9473684211 & -8.89 x_{3} & -0.95 x_{9} & -14.58 x_{2} & -0.68 x_{12} & -11.37 x_{4} & -10.95 x_{6} & + 12.68 x_{1} & -18.95 x_{8}\\
 x_{7}   &  5.73684210526 & -0.14 x_{3} & -0.07 x_{9} & -0.89 x_{2} & -0.09 x_{12} & -0.84 x_{4} & -0.40 x_{6} & +  0.75 x_{1} & -1.40 x_{8}\\
 x_{14}   &  34.0 & -7.33 x_{3} & -0.67 x_{9} & +  2.00 x_{2} & -0.33 x_{12} & -9.00 x_{4} & -5.33 x_{6} & + 12.67 x_{1} & -18.33 x_{8}\\
\hline
z    &  28.9210526316 & -0.68 x_{3} & -0.59 x_{9} & -6.87 x_{2} & -0.36 x_{12} & -2.55 x_{4} & -1.25 x_{6} & +  0.19 x_{1} & -8.75 x_{8}\\
\end{array}\]


 $ x_{1} $ enters and Unbounded Dictionary!


 $ x_{1} $ enters and Unbounded Dictionary!
\end{document}
